question_answer

The area of the triangle formed by the coordinate axes and the normal to the curve \[y={{e}^{2x}}+{{x}^{2}}\]at the point

A) (0, 1) is  \[0\]                    

B)         \[1\,sq\,unit\]

C)  \[\frac{1}{2}\,sq\,unit\]                              

D)  \[2\,sq\,unit\]

E)  \[4\,sq\,unit\]

Correct Answer: B

Solution :

Given curve is\[y={{e}^{2x}}+{{x}^{2}}\] \[\therefore \]  \[\frac{dy}{dx}=2{{e}^{2x}}+2x\] \[\Rightarrow \]               \[{{\left( \frac{dy}{dx} \right)}_{(0,1)}}=2\] \[\therefore \]Equation of normal is                 \[y-1=-\frac{1}{2}(x-0)\] \[\Rightarrow \]               \[y+\frac{x}{2}=1\] \[\therefore \]Area of \[\Delta OAB=\frac{1}{2}\times 2\times 1\] \[=1\text{ }sq\text{ }unit\]